:: by Wojciech A. Trybulec

::

:: Received July 27, 1990

:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let GF be non empty doubleLoopStr ;

let V be non empty ModuleStr over GF;

let IT be Subset of V;

end;
let V be non empty ModuleStr over GF;

let IT be Subset of V;

attr IT is linearly-independent means :: VECTSP_7:def 1

for l being Linear_Combination of IT st Sum l = 0. V holds

Carrier l = {} ;

for l being Linear_Combination of IT st Sum l = 0. V holds

Carrier l = {} ;

:: deftheorem defines linearly-independent VECTSP_7:def 1 :

for GF being non empty doubleLoopStr

for V being non empty ModuleStr over GF

for IT being Subset of V holds

( IT is linearly-independent iff for l being Linear_Combination of IT st Sum l = 0. V holds

Carrier l = {} );

for GF being non empty doubleLoopStr

for V being non empty ModuleStr over GF

for IT being Subset of V holds

( IT is linearly-independent iff for l being Linear_Combination of IT st Sum l = 0. V holds

Carrier l = {} );

notation

let GF be non empty doubleLoopStr ;

let V be non empty ModuleStr over GF;

let IT be Subset of V;

antonym linearly-dependent IT for linearly-independent ;

end;
let V be non empty ModuleStr over GF;

let IT be Subset of V;

antonym linearly-dependent IT for linearly-independent ;

theorem :: VECTSP_7:1

for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for A, B being Subset of V st A c= B & B is linearly-independent holds

A is linearly-independent

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for A, B being Subset of V st A c= B & B is linearly-independent holds

A is linearly-independent

proof end;

theorem Th2: :: VECTSP_7:2

for R being non degenerated Ring

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

not 0. V in A

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

not 0. V in A

proof end;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr ;

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;

coherence

for b_{1} being Subset of V st b_{1} is empty holds

b_{1} is linearly-independent

end;
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;

coherence

for b

b

proof end;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr ;

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;

existence

ex b_{1} being Subset of V st

( b_{1} is finite & b_{1} is linearly-independent )

end;
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;

existence

ex b

( b

proof end;

theorem :: VECTSP_7:3

for R being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being LeftMod of R

for v being Vector of V holds

( {v} is linearly-independent iff v <> 0. V )

for V being LeftMod of R

for v being Vector of V holds

( {v} is linearly-independent iff v <> 0. V )

proof end;

theorem Th4: :: VECTSP_7:4

for GF being non empty non degenerated right_complementable well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds

v1 <> 0. V

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds

v1 <> 0. V

proof end;

theorem Th5: :: VECTSP_7:5

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v1, v2 being Vector of V holds

( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a being Element of GF holds v1 <> a * v2 ) ) )

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v1, v2 being Vector of V holds

( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a being Element of GF holds v1 <> a * v2 ) ) )

proof end;

theorem :: VECTSP_7:6

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v1, v2 being Vector of V holds

( ( v1 <> v2 & {v1,v2} is linearly-independent ) iff for a, b being Element of GF st (a * v1) + (b * v2) = 0. V holds

( a = 0. GF & b = 0. GF ) )

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v1, v2 being Vector of V holds

( ( v1 <> v2 & {v1,v2} is linearly-independent ) iff for a, b being Element of GF st (a * v1) + (b * v2) = 0. V holds

( a = 0. GF & b = 0. GF ) )

proof end;

TH2: for GF being Ring

for V being LeftMod of GF

for L being Linear_Combination of V

for C being finite Subset of V st Carrier L c= C holds

ex F being FinSequence of V st

( F is one-to-one & rng F = C & Sum L = Sum (L (#) F) )

proof end;

TH3: for GF being Ring

for V being LeftMod of GF

for L being Linear_Combination of V

for a being Scalar of holds Sum (a * L) = a * (Sum L)

proof end;

definition

let GF be Ring;

let V be LeftMod of GF;

let A be Subset of V;

ex b_{1} being strict Subspace of V st the carrier of b_{1} = { (Sum l) where l is Linear_Combination of A : verum }

for b_{1}, b_{2} being strict Subspace of V st the carrier of b_{1} = { (Sum l) where l is Linear_Combination of A : verum } & the carrier of b_{2} = { (Sum l) where l is Linear_Combination of A : verum } holds

b_{1} = b_{2}
by VECTSP_4:29;

end;
let V be LeftMod of GF;

let A be Subset of V;

func Lin A -> strict Subspace of V means :Def2: :: VECTSP_7:def 2

the carrier of it = { (Sum l) where l is Linear_Combination of A : verum } ;

existence the carrier of it = { (Sum l) where l is Linear_Combination of A : verum } ;

ex b

proof end;

uniqueness for b

b

:: deftheorem Def2 defines Lin VECTSP_7:def 2 :

for GF being Ring

for V being LeftMod of GF

for A being Subset of V

for b_{4} being strict Subspace of V holds

( b_{4} = Lin A iff the carrier of b_{4} = { (Sum l) where l is Linear_Combination of A : verum } );

for GF being Ring

for V being LeftMod of GF

for A being Subset of V

for b

( b

theorem Th7: :: VECTSP_7:7

for x being object

for GF being Ring

for V being LeftMod of GF

for A being Subset of V holds

( x in Lin A iff ex l being Linear_Combination of A st x = Sum l )

for GF being Ring

for V being LeftMod of GF

for A being Subset of V holds

( x in Lin A iff ex l being Linear_Combination of A st x = Sum l )

proof end;

theorem Th8: :: VECTSP_7:8

for x being object

for GF being Ring

for V being LeftMod of GF

for A being Subset of V st x in A holds

x in Lin A

for GF being Ring

for V being LeftMod of GF

for A being Subset of V st x in A holds

x in Lin A

proof end;

theorem :: VECTSP_7:10

for GF being Ring

for V being LeftMod of GF

for A being Subset of V holds

( not Lin A = (0). V or A = {} or A = {(0. V)} )

for V being LeftMod of GF

for A being Subset of V holds

( not Lin A = (0). V or A = {} or A = {(0. V)} )

proof end;

theorem Th11: :: VECTSP_7:11

for GF being non degenerated Ring

for V being LeftMod of GF

for A being Subset of V

for W being strict Subspace of V st A = the carrier of W holds

Lin A = W

for V being LeftMod of GF

for A being Subset of V

for W being strict Subspace of V st A = the carrier of W holds

Lin A = W

proof end;

theorem :: VECTSP_7:12

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being strict VectSp of GF

for A being Subset of V st A = the carrier of V holds

Lin A = V

for V being strict VectSp of GF

for A being Subset of V st A = the carrier of V holds

Lin A = V

proof end;

theorem Th13: :: VECTSP_7:13

for GF being Ring

for V being LeftMod of GF

for A, B being Subset of V st A c= B holds

Lin A is Subspace of Lin B

for V being LeftMod of GF

for A, B being Subset of V st A c= B holds

Lin A is Subspace of Lin B

proof end;

theorem :: VECTSP_7:14

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for A, B being Subset of V st Lin A = V & A c= B holds

Lin B = V

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for A, B being Subset of V st Lin A = V & A c= B holds

Lin B = V

proof end;

theorem :: VECTSP_7:15

for GF being Ring

for V being LeftMod of GF

for A, B being Subset of V holds Lin (A \/ B) = (Lin A) + (Lin B)

for V being LeftMod of GF

for A, B being Subset of V holds Lin (A \/ B) = (Lin A) + (Lin B)

proof end;

theorem :: VECTSP_7:16

for GF being Ring

for V being LeftMod of GF

for A, B being Subset of V holds Lin (A /\ B) is Subspace of (Lin A) /\ (Lin B)

for V being LeftMod of GF

for A, B being Subset of V holds Lin (A /\ B) is Subspace of (Lin A) /\ (Lin B)

proof end;

definition

let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

let V be LeftMod of GF;

let A be Subset of V;

end;
let V be LeftMod of GF;

let A be Subset of V;

:: deftheorem Def3 defines base VECTSP_7:def 3 :

for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

for V being LeftMod of GF

for A being Subset of V holds

( A is base iff ( A is linearly-independent & Lin A = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ) );

for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

for V being LeftMod of GF

for A being Subset of V holds

( A is base iff ( A is linearly-independent & Lin A = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ) );

Lm1: for R being Ring

for a being Scalar of R st - a = 0. R holds

a = 0. R

proof end;

Lm2: for R being non degenerated almost_left_invertible Ring

for V being LeftMod of R

for a being Scalar of

for v being Vector of V st a <> 0. R holds

( (a ") * (a * v) = (1. R) * v & ((a ") * a) * v = (1. R) * v )

proof end;

theorem Th17: :: VECTSP_7:17

for R being non degenerated almost_left_invertible Ring

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

ex B being Subset of V st

( A c= B & B is base )

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

ex B being Subset of V st

( A c= B & B is base )

proof end;

theorem Th18: :: VECTSP_7:18

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for A being Subset of V st Lin A = V holds

ex B being Subset of V st

( B c= A & B is linearly-independent & Lin B = V )

for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for A being Subset of V st Lin A = V holds

ex B being Subset of V st

( B c= A & B is linearly-independent & Lin B = V )

proof end;

Lem0A: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

for V being LeftMod of GF holds {} the carrier of V is linearly-independent

proof end;

registration

let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

let V be LeftMod of GF;

coherence

{} the carrier of V is linearly-independent by Lem0A;

end;
let V be LeftMod of GF;

coherence

{} the carrier of V is linearly-independent by Lem0A;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

let V be LeftMod of GF;

existence

ex b_{1} being Subset of V st b_{1} is base

end;
let V be LeftMod of GF;

existence

ex b

proof end;

definition

let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

let IT be LeftMod of GF;

end;
let IT be LeftMod of GF;

:: deftheorem defines free VECTSP_7:def 4 :

for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

for IT being LeftMod of GF holds

( IT is free iff ex B being Subset of IT st B is base );

for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

for IT being LeftMod of GF holds

( IT is free iff ex B being Subset of IT st B is base );

Lem1A: for GF being non empty right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

for V being LeftMod of GF holds (0). V is free

proof end;

registration

let GF be non empty right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

let V be LeftMod of GF;

coherence

(0). V is free by Lem1A;

end;
let V be LeftMod of GF;

coherence

(0). V is free by Lem1A;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

ex b_{1} being LeftMod of GF st

( b_{1} is strict & b_{1} is free )

end;
cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed free for ModuleStr over GF;

existence ex b

( b

proof end;

definition

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

let V be LeftMod of GF;

mode Basis of V is base Subset of V;

end;
let V be LeftMod of GF;

mode Basis of V is base Subset of V;

theorem :: VECTSP_7:19

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being LeftMod of GF

for A being Subset of V st A is linearly-independent holds

ex I being Basis of V st A c= I

for V being LeftMod of GF

for A being Subset of V st A is linearly-independent holds

ex I being Basis of V st A c= I

proof end;

theorem :: VECTSP_7:20

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr

for V being LeftMod of GF

for A being Subset of V st Lin A = V holds

ex I being Basis of V st I c= A

for V being LeftMod of GF

for A being Subset of V st Lin A = V holds

ex I being Basis of V st I c= A

proof end;

:: Additional, copied from MOD_3

theorem :: VECTSP_7:21

theorem :: VECTSP_7:22

theorem :: VECTSP_7:23

for R being non degenerated Ring

for V being LeftMod of R

for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds

( v1 <> 0. V & v2 <> 0. V )

for V being LeftMod of R

for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds

( v1 <> 0. V & v2 <> 0. V )

proof end;

theorem :: VECTSP_7:24

for R being domRing

for V being LeftMod of R

for L being Linear_Combination of V

for a being Scalar of R st a <> 0. R holds

Carrier (a * L) = Carrier L

for V being LeftMod of R

for L being Linear_Combination of V

for a being Scalar of R st a <> 0. R holds

Carrier (a * L) = Carrier L

proof end;

theorem Th12: :: VECTSP_7:25

for R being domRing

for V being LeftMod of R

for A being Subset of V

for W being strict Subspace of V st not R is degenerated & A = the carrier of W holds

Lin A = W

for V being LeftMod of R

for A being Subset of V

for W being strict Subspace of V st not R is degenerated & A = the carrier of W holds

Lin A = W

proof end;

theorem :: VECTSP_7:26

for R being domRing

for V being strict LeftMod of R

for A being Subset of V st not R is degenerated & A = the carrier of V holds

Lin A = V

for V being strict LeftMod of R

for A being Subset of V st not R is degenerated & A = the carrier of V holds

Lin A = V

proof end;

theorem :: VECTSP_7:27

for R being domRing

for V being strict LeftMod of R

for A, B being Subset of V st Lin A = V & A c= B holds

Lin B = V

for V being strict LeftMod of R

for A, B being Subset of V st Lin A = V & A c= B holds

Lin B = V

proof end;

theorem :: VECTSP_7:28

for R being Ring

for V being LeftMod of R

for v1, v2 being Vector of V st not R is degenerated & {v1,v2} is linearly-independent holds

( v1 <> 0. V & v2 <> 0. V )

for V being LeftMod of R

for v1, v2 being Vector of V st not R is degenerated & {v1,v2} is linearly-independent holds

( v1 <> 0. V & v2 <> 0. V )

proof end;